The introduction of the new variable \(\xi \)  and additional constraints preserves the equivalence between (REF ) and (REF ): from a solution of one problem, a solution of the other is readily found, and vice versa [1]}. Similar to (REF ), problem (REF ) also can be solved using bilevel optimization. However, the formulation of (REF ) permits only keeping the equality constraints in the lower-level optimization:
\(\begin{aligned}& \underset{\sigma }{\text{minimize}} && J(\sigma ,T)\\& \text{subject to} && C(T) \sigma  = \xi ,\\& && A(T) \sigma  = b,\end{aligned}\qquad \mathrm {(LLP)}\) 
